Page 227 -

210 5. Some worked examples arising from physical problems

with

where

One essential difficulty that we have overlooked thus far is that the process of heating
the material, starting from ambient conditions at t = 0, requires the temperature to be
raised before any vaporisation occurs, and therefore before the hole can begin to form.
Because we have a failure of our original asymptotic expansion only when
this should contain the time period over which the initial heating phase occurs. Let the
hole be unformed for (and we assume that if it is smaller, we
will rescale), then in this time interval we have and so the boundary
condition (5.35) now reads

The resulting problem for (which excludes (5.34c) and uses (5.36) in place of (5.35))
no longer contains and it can be solved exactly:

(Many of these standard solutions that we use in theories of heat conduction can be
found in Carslaw & Jaeger, 1959.) The additional boundary condition (5.35c) now
becomes the condition which, when attained, heralds the formation of the hole. From
(5.37), we see that on at the time

and after this time the hole develops.
Finally, we complete the formulation of the problem for but for times
larger than we introduce and write
with to give

with

and